05 - Simplify Irrational Exponents, Part 1 (Radical Exponents, Powers, Pi & More) - By Math and Science
Transcript
| 00:00 | Hello . Welcome back to algebra . The title of | |
| 00:02 | this lesson is called irrational exponents . Now it's a | |
| 00:05 | complicated sounding title but I promise you by the individual | |
| 00:08 | understand exactly what an irrational exponent is , how to | |
| 00:11 | deal with them and why they're important . I'm also | |
| 00:13 | really excited to teach this lesson because In the course | |
| 00:16 | of this lesson today right now you're going to learn | |
| 00:19 | the very first concepts that you learn in calculus one | |
| 00:22 | . Now I very firmly believe that everything in math | |
| 00:27 | can be fully understandable by everybody and I firmly believe | |
| 00:31 | in not babying people too much . So what I'm | |
| 00:33 | doing here is we're gonna introduce and use the concept | |
| 00:35 | of irrational exponents . But in the course of the | |
| 00:37 | lesson you're going to understand logically why that idea , | |
| 00:41 | that very first idea and calculus one that everyone learns | |
| 00:44 | is very applicable to this . So pat yourself on | |
| 00:47 | the back when you're done with this lesson you're going | |
| 00:49 | to understand the very very fundamental concept of calculus one | |
| 00:52 | that everyone learns the very first day of calculus one | |
| 00:55 | in the course of this algebra lesson . So let's | |
| 00:57 | crawl before we can walk . I promise it will | |
| 00:59 | all be understandable . We want to talk about irrational | |
| 01:02 | exponents , those are exponents that are just for lack | |
| 01:06 | of a better word . Weird . They cannot be | |
| 01:07 | written as fractions in irrational exponent would be like uh | |
| 01:11 | two to the power of pie because pies in a | |
| 01:14 | rational number cannot be written as a fraction or two | |
| 01:16 | to the power of square root of three . Square | |
| 01:19 | root of three is irrational . It cannot be written | |
| 01:20 | as a fraction . So how do you calculate things | |
| 01:23 | that have radicals or pie or other weird things in | |
| 01:27 | the expanded itself ? How do you handle it ? | |
| 01:29 | So let's before we do that , let's take a | |
| 01:31 | little bit of a trip down memory lane and talk | |
| 01:33 | about what we know . First of all , we | |
| 01:35 | already know how to deal with positive exponents , Positive | |
| 01:38 | expose we've been dealing about for a long time . | |
| 01:41 | For instance , to to the positive three power Is | |
| 01:44 | defined as two times two times two , which is | |
| 01:47 | eight . So we know how to deal with positive | |
| 01:48 | experience . We also have learned how to deal extensively | |
| 01:52 | with what have what you do when you have a | |
| 01:54 | negative exponents . So for instance , we know how | |
| 01:57 | to deal with it when we have to to the | |
| 01:59 | negative fourth power . What we do is we take | |
| 02:01 | this negative exponent and we bring it to the bottom | |
| 02:03 | of the fraction and make it a positive exponents . | |
| 02:06 | That negative means you drag it downstairs . Now what | |
| 02:08 | you have down here is 1/16 because two times two | |
| 02:12 | times two times two will be 16 . So we | |
| 02:14 | know how to deal with negative exponents . We have | |
| 02:17 | also talked extensively about what happens when the exponent is | |
| 02:20 | actually the number zero . It turns out that anything | |
| 02:23 | raised to the zero power is defined mathematically just to | |
| 02:27 | be the number one . And we've talked about in | |
| 02:29 | the past why this is the case . So if | |
| 02:31 | you're curious why it's that way , go back to | |
| 02:33 | my very first lessons on exponents and I talked about | |
| 02:35 | at great length and then finally very recently the last | |
| 02:39 | lesson we have talked about what to do when you | |
| 02:41 | have a rational exponents , that means you have an | |
| 02:43 | exponent that can be written as a fraction . So | |
| 02:45 | for instance if you have to to the one third | |
| 02:47 | power , This is just defined to be the cube | |
| 02:52 | root of the # two . Okay , or another | |
| 02:55 | example would be a little more complicated example in this | |
| 02:59 | would be what do you do if you have four | |
| 03:01 | to the 3/4 power . Right ? We can write | |
| 03:05 | this four to the 3/4 power . A couple of | |
| 03:07 | different ways . We can write it as uh four | |
| 03:10 | raised to the third power and then to the fourth | |
| 03:14 | to the 1/4 power . Because we can multiply these | |
| 03:16 | powers together to give us this . And then when | |
| 03:19 | we write it this way then what we have is | |
| 03:22 | four to the third power . We write it as | |
| 03:25 | the fourth root of that . But another way to | |
| 03:28 | write it how switch colors to show you that is | |
| 03:30 | instead of writing it like this , you could say | |
| 03:32 | that four to the third to the three . Fourth | |
| 03:35 | power can be writing this four to the 1/4 power | |
| 03:38 | . And we raise that quantity to the third party | |
| 03:40 | . See the the order here can be interchanged because | |
| 03:43 | when you multiply this and you multiply this you get | |
| 03:45 | exactly the same thing . So if we do it | |
| 03:47 | this way we have a four that we take the | |
| 03:50 | fourth root of and then we raise that quantity to | |
| 03:52 | the third power . So if we have a fractional | |
| 03:55 | exponent like this we can raise , let me just | |
| 03:57 | check myself as 434 to the third power and then | |
| 04:00 | the fourth root of that . Or we can write | |
| 04:02 | it as the four , doing the fourth root first | |
| 04:04 | and then raising it to the third power . So | |
| 04:07 | we know we have to raise it to a power | |
| 04:09 | and then take the route . We can just do | |
| 04:10 | them in whatever order we want . Now this is | |
| 04:12 | all stuff that we have learned in the past . | |
| 04:15 | What do you do when you have a positive exponent | |
| 04:17 | ? What do you do when you have a negative | |
| 04:18 | exponent ? What if you do when you have a | |
| 04:19 | zero exponent ? What do you do when you have | |
| 04:21 | a fractional exponent ? We've covered all of that . | |
| 04:23 | Now what we need to do is what does this | |
| 04:26 | mean ? What does this mean ? And what am | |
| 04:35 | I referring to just as an example , what if | |
| 04:37 | you have two to the power of square root of | |
| 04:39 | three ? We have never covered that before . We've | |
| 04:42 | never talked about . What do you do when the | |
| 04:43 | exponent itself ? It cannot be written as a fraction | |
| 04:46 | or a negative or a positive number or zero . | |
| 04:48 | What do you do ? Well the punch line , | |
| 04:50 | I'm gonna give you the punchline and then we're gonna | |
| 04:51 | go and do a little bit of background work to | |
| 04:53 | kind of show you why it's the case the punch | |
| 04:55 | line is . You treat these exponents As exactly what | |
| 05:00 | the same rules as all of the other exponents . | |
| 05:02 | You can add them , you can subtract them , | |
| 05:04 | you can multiply them . It depends on the situation | |
| 05:06 | you're in . We're going to do a lot of | |
| 05:07 | problems like that , but this radical appear should not | |
| 05:10 | scare you . Right ? You're going to use the | |
| 05:12 | same rules of algebra . What if you have three | |
| 05:14 | to the power of you know , two pi Because | |
| 05:17 | Pi is irrational . You it's an exponents a number | |
| 05:20 | 3.14159 pie goes on and on and on forever . | |
| 05:23 | It's irrational . But you treat uh those exponents with | |
| 05:27 | exactly the same rules that you treat any other experts | |
| 05:30 | . Let's explore why that is the case . And | |
| 05:32 | in the course of learning how to calculate that and | |
| 05:36 | what it really means to calculate something like this . | |
| 05:38 | We're gonna learn a tiny bit of calculus right along | |
| 05:40 | the way so that we're going to do here . | |
| 05:42 | How would we calculate two to the power of square | |
| 05:45 | root of 3 ? All right . Two to the | |
| 05:47 | power of square root of three really is the following | |
| 05:51 | thing . Now I know we're not all human calculators | |
| 05:54 | , but basically the square root of three is when | |
| 05:57 | you truncated is 1.732 dot dot dot dot dot dot | |
| 06:02 | . That means these decimals go on and on and | |
| 06:04 | on . They never repeat and they're not because they | |
| 06:07 | don't repeat . You can't write it as a fraction | |
| 06:09 | . That's what it means to be irrational . So | |
| 06:11 | how do you calculate an expo that has an infinite | |
| 06:13 | set of decimals that go on and on and on | |
| 06:15 | and on like that ? Well , you have to | |
| 06:17 | use a concept from calculus . That's very very very | |
| 06:21 | uh ties very much into what we're doing here . | |
| 06:23 | So we're gonna kind of go over that . What | |
| 06:25 | you do is you forget about the all the decimal | |
| 06:27 | places way over here . What you do is you | |
| 06:28 | say I want to approximate this . Let's approximate since | |
| 06:31 | it's 1.7 roughly speaking then let's first approximation , let's | |
| 06:36 | just say it's to the first power . That's a | |
| 06:38 | really bad approximation of this actual number . But if | |
| 06:41 | you would take it to the first power , you | |
| 06:43 | would just get an exact number of two . All | |
| 06:45 | right , let's make a slightly better approximation . Let's | |
| 06:48 | say it's two to the power of 1.7 . That | |
| 06:51 | is a little bit closer to the actual value of | |
| 06:53 | the square to three , but it's obviously not quite | |
| 06:55 | right because these decimals go on and on forever , | |
| 06:58 | But this 1.7 how do you evaluate a decimal like | |
| 07:00 | that ? We'll see any decimal that doesn't go on | |
| 07:03 | and on forever . Like this one I can write | |
| 07:05 | it as a fraction , I can say this is | |
| 07:06 | 17/10 . I want you to convince yourself that 17/10 | |
| 07:11 | is exactly the same thing as 1.7 because when you | |
| 07:14 | divide by 10 you take the decimal and move at | |
| 07:16 | one spot to the left . So 1.7 is exactly | |
| 07:19 | 17 10th and you now know how to evaluate exponents | |
| 07:24 | with fractions , it's really gnarly but you can do | |
| 07:28 | it , this is to raise to the 17th power | |
| 07:31 | , Take the result of that and you take the | |
| 07:32 | 10th root of it . Or you could say to | |
| 07:35 | take the 10th root of it and then raise the | |
| 07:37 | result to the 17th power . So you can calculate | |
| 07:40 | this exactly And if you did that you would get | |
| 07:44 | 3.2490 . So this number is a little bit closer | |
| 07:51 | to the actual square root of three . And as | |
| 07:53 | a result , this number that you get is a | |
| 07:55 | little bit closer than our first approximation , which is | |
| 07:57 | really bad because this was really not close at all | |
| 08:00 | , but it was where we started from . Okay | |
| 08:02 | , alright , let's switch colors and let's get a | |
| 08:04 | little bit closer , let's get a little closer to | |
| 08:06 | the square root of 32.1 point 73 We took one | |
| 08:11 | more of the decimals here . How would we calculate | |
| 08:14 | this ? It would be to to the 173 divided | |
| 08:18 | by 100 . Now this is again a gnarly , | |
| 08:20 | gnarly , Very ugly uh fractional exponent . But again | |
| 08:24 | , two to the 173 power , you can calculate | |
| 08:27 | that , get the result and you can take the | |
| 08:29 | 100th root of it , you can do that , | |
| 08:31 | you can do a giant factor tree , look for | |
| 08:33 | a set of 100 commonality , you can get the | |
| 08:36 | exact number but when you run it through a calculator | |
| 08:38 | , the number that you're gonna get is three 3173 | |
| 08:43 | . You see what's happening ? Our first approximation was | |
| 08:46 | very crude . But as we get a little closer | |
| 08:48 | we get a little closer to the real value . | |
| 08:50 | As we get slightly more closer we get even a | |
| 08:52 | little bit closer to the value . Let's give it | |
| 08:54 | one more time . Let's go one more time to | |
| 08:57 | get a little closer to that . 1.732 . We | |
| 09:00 | take one more decimal . Of course there's more decimals | |
| 09:03 | after it . But this is going to be how | |
| 09:05 | would we calculated 2 to the power of 1732 divided | |
| 09:09 | by 1000 . Because you divide by 1000 you move | |
| 09:12 | the decimal three places 123 that's exactly what it is | |
| 09:16 | . So to raise to the power of 1732 that's | |
| 09:20 | crazy . Take the value and then you take the | |
| 09:22 | 1000 through the computer can do that . I'm not | |
| 09:25 | gonna do that . What you're gonna get is the | |
| 09:27 | value exactly 3.321 night . So the whole point of | |
| 09:32 | this exercise is to show you people say how do | |
| 09:34 | you calculate an exponent that has a radical in it | |
| 09:37 | like that . That's irrational because the decimals never end | |
| 09:40 | . The truth is you can't calculate it exactly right | |
| 09:43 | by by because of the decimals never really end . | |
| 09:46 | But what you can do is you can get closer | |
| 09:48 | and closer and closer and closer By taking more and | |
| 09:52 | more and more of the decimals and drawing in a | |
| 09:54 | closer approximation to the number and you can see that | |
| 09:57 | my answer is getting closer and closer to a real | |
| 10:00 | value notice this is 3.32 and some change this is | |
| 10:03 | 3.3 and some change this is .3.3 and some change | |
| 10:06 | , it's getting closer and closer to a final value | |
| 10:09 | . 3.3 something something it's gonna be the actual value | |
| 10:13 | of two to the power of square to three . | |
| 10:14 | So if I continue this process then ultimately I want | |
| 10:20 | to calculate two to the power of square 23 And | |
| 10:22 | I'm getting closer and closer and closer , this is | |
| 10:24 | the exact number I want and when you run this | |
| 10:27 | through a calculator or computer , the actual number you're | |
| 10:29 | gonna get is 3.3219971 dot dot dot . The reason | |
| 10:37 | the dots is because this is an irrational exponents , | |
| 10:40 | it has infinite decimal places in the exponents but I | |
| 10:42 | could of course take more and more and more decimals | |
| 10:44 | and I would get an approximation closer and closer and | |
| 10:46 | closer to the uh to the answer you get from | |
| 10:50 | a computer , which would be something like this . | |
| 10:52 | You see the approximation is getting closer and closer so | |
| 10:55 | what you can do and this is the part where | |
| 10:57 | the kind of the calculus comes into it here , | |
| 10:59 | you can define to find notice . This is we | |
| 11:05 | took a successive approximations to to the first two to | |
| 11:08 | the 1.72 to the 1.732 to the 1.73 to 2 | |
| 11:12 | to the this is the exact thing we're trying to | |
| 11:14 | find . You can define something called two to the | |
| 11:17 | power of x . Notice by putting a variable in | |
| 11:20 | the exponents , I can actually change the value of | |
| 11:23 | the exponents just like I did here , getting closer | |
| 11:25 | and closer and closer . I'm just changing the exponent | |
| 11:27 | here by putting the variable in the exponents . It | |
| 11:30 | allows me to change it to whatever I want because | |
| 11:32 | it's a variable up there . Okay , this thing | |
| 11:36 | to to the value of X is called an exponential | |
| 11:42 | function . Now , in just a little bit we | |
| 11:48 | are going to study exponential functions in great detail . | |
| 11:51 | It's one of the most important functions in all of | |
| 11:53 | math and there's tons of reasons I can give you | |
| 11:56 | give you for that to to tell you that so | |
| 11:58 | you believe me but I'm not gonna do that now | |
| 12:00 | right now . We're just focusing on understanding this rational | |
| 12:03 | expo but just know this thing is called an exponential | |
| 12:05 | function . We're going to learn all about it right | |
| 12:08 | now , if you let X approach the value of | |
| 12:14 | the square root of three , that means he gets | |
| 12:16 | closer and closer and closer , just like I did | |
| 12:17 | here , closer , closer , closer closer , I | |
| 12:20 | can't really ever get here because it's infinite decimal places | |
| 12:23 | but I can get closer and closer and closer if | |
| 12:25 | I let it get closer and closer and closer , | |
| 12:27 | uh I'll be right over here closer and closer . | |
| 12:32 | I mean I can continue writing decimals to the end | |
| 12:34 | of the universe but I'm still never gonna quite get | |
| 12:36 | there but I'll get infinitely close to this guy . | |
| 12:39 | Then what you can write down is the following very | |
| 12:42 | , very important thing . You can take what we | |
| 12:46 | call the limit as X approaches the number , the | |
| 12:50 | square root of three , but it never quite really | |
| 12:52 | gets there . But it gets really close and I'm | |
| 12:55 | gonna take the limit of the function to to the | |
| 12:57 | power of X . That means I put an X | |
| 13:00 | value in a really crude one of one and then | |
| 13:02 | I put an X value and a little bit closer | |
| 13:04 | . 1.7 and then I put a little bit closer | |
| 13:06 | . 1.73 A little bit closer . 1.732 A little | |
| 13:09 | bit closer . A little bit closer . A little | |
| 13:10 | bit closer . I get infinitely close to this very | |
| 13:13 | specific thing I'm trying to calculate then what's going to | |
| 13:16 | happen is I'm going to then have a answer which | |
| 13:21 | is approximately because I can never really quite get there | |
| 13:23 | of 3.321971 dot dot dot dot dot never ends . | |
| 13:29 | So all of this to tell you that the way | |
| 13:33 | that you actually calculate fractional exponents , like if you | |
| 13:36 | really had to do it by paper and pencil , | |
| 13:37 | I mean we all know we can go to a | |
| 13:39 | calculator and punching in there and get the , get | |
| 13:41 | the thing but with the calculator is doing is it | |
| 13:43 | takes successive approximations really , really close and to as | |
| 13:47 | many decimals as your display can show you and it | |
| 13:49 | just gives you that guy truncate the rest , it | |
| 13:52 | doesn't do it all the way . But mathematically you | |
| 13:55 | can take this thing called a limit which means we | |
| 13:58 | define a new function which is called exponential function where | |
| 14:01 | the variables up top in the exponent and we let | |
| 14:04 | this variable get closer and closer and closer and closer | |
| 14:06 | to what we're really trying to calculate two to the | |
| 14:08 | power of the squared of three and then we're going | |
| 14:11 | to get However we get as far as close as | |
| 14:13 | we can see the real actual value . You can | |
| 14:15 | get a computer to calculate this thing to 100 million | |
| 14:17 | decimal places or 500 trillion decimal places . It's just | |
| 14:21 | gonna take time . But mathematically this thing called a | |
| 14:23 | limit is Is writing down what you're doing , you're | |
| 14:26 | getting closer and closer to calculate that value . This | |
| 14:29 | concept of the limit is the very first thing you | |
| 14:31 | learn in calculus one . It is really the foundation | |
| 14:34 | upon which all of calculus is based . So you've | |
| 14:37 | now learned it in general idea here in algebra . | |
| 14:41 | All right , so now that we know what a | |
| 14:45 | irrational exponent is , it's an exponent that has uh | |
| 14:49 | can be written as a fraction . Mathematically we would | |
| 14:52 | get closer and closer to the actual value by doing | |
| 14:54 | something like this . This is what a computer does | |
| 14:57 | . Calculus defines what these things are by taking this | |
| 15:00 | thing called a limit . As the exponent of this | |
| 15:03 | thing called an exponential function gets closer and closer now | |
| 15:06 | , what we need to do is solve a couple | |
| 15:08 | of problems to show you how to handle it if | |
| 15:11 | you actually have an irrational exponent there . So for | |
| 15:14 | instance if you have three to the power of the | |
| 15:16 | square root of two , multiplied by three to the | |
| 15:19 | power of square root of two . How do you | |
| 15:21 | handle it ? Obviously I don't want to calculate all | |
| 15:24 | the decimal glory details of what that power is . | |
| 15:27 | I want to keep it exact . I want to | |
| 15:29 | keep it as a radical but you follow the same | |
| 15:31 | rules of algebra to get there . So here you | |
| 15:34 | have three to a power three to a power the | |
| 15:36 | basis of the same . So you can add these | |
| 15:38 | exponents together Square root of two plus the square root | |
| 15:42 | of two . How do I know I can do | |
| 15:44 | that ? I want you to think this is exactly | |
| 15:50 | the same as X squared times X cubed the basis | |
| 15:55 | of the same . I add the exponents base is | |
| 15:58 | the same , add the exponents . But in this | |
| 16:00 | case the exponents are just really weird . So what | |
| 16:03 | I have is when I add squared of two plus | |
| 16:05 | the square root of two , it's just two times | |
| 16:08 | the square root of two . Because I've added them | |
| 16:10 | together . There's two of them there . Now I | |
| 16:11 | can leave this as the correct as the final answer | |
| 16:14 | . I can do that but I want to go | |
| 16:16 | a little step further and try to simplify it . | |
| 16:18 | If I can I can write this as three squared | |
| 16:22 | raised to the square root of two . How can | |
| 16:24 | I do that ? Because I multiply when I have | |
| 16:26 | a power raised to a power I multiply them . | |
| 16:28 | That gives me what I have here . But I | |
| 16:30 | know that three squared is nine To the race of | |
| 16:33 | the power of the squared of two . Okay , | |
| 16:36 | and that's the final answer . Now do not try | |
| 16:38 | . This does not mean that you're taking the square | |
| 16:41 | root of nine . It doesn't mean that the radical | |
| 16:43 | applies to the nine , it means that it's the | |
| 16:45 | number nine raised to a power but that power is | |
| 16:49 | not written as a fraction . It can't be because | |
| 16:51 | this square root of two is irrational . If you | |
| 16:54 | put it in a calculator you're gonna get all these | |
| 16:56 | decimals that go on and on forever because it cannot | |
| 16:59 | be written as one half or one third or 1/4 | |
| 17:01 | or 1/5 or 1/8 or whatever . It can't it's | |
| 17:03 | just a squared of two . That's the exact value | |
| 17:06 | . So you leave your answer as nine raised to | |
| 17:08 | the power of this radical notice we use the same | |
| 17:11 | rules of algebra , we just added them together . | |
| 17:13 | Now if you really had to calculate the decimal form | |
| 17:16 | of it , you would do something like this , | |
| 17:18 | you would start with a crude approximation . Get closer | |
| 17:20 | , get closer , get closer and you would get | |
| 17:22 | as many decimals as you needed to calculate approximation for | |
| 17:25 | the answer . But the real answer would be this | |
| 17:29 | limit concept . You would take nine to the power | |
| 17:32 | of X and you would get X get closer and | |
| 17:34 | closer and closer to the square root of two . | |
| 17:36 | Getting more and more and more accuracy . And then | |
| 17:38 | in the in the infinite limit there as you get | |
| 17:40 | closer and closer to it , that would be what | |
| 17:41 | the value actually is . All right . What if | |
| 17:44 | we have four raised to the power of the square | |
| 17:47 | root of two and then we're gonna raise that entire | |
| 17:50 | thing . That's the square of to their to the | |
| 17:53 | third power . How do we handle that ? Well | |
| 17:55 | , it's the same rules of algebra . We have | |
| 17:57 | a power raised to a power . So we multiply | |
| 18:00 | those together four times and then three times the square | |
| 18:04 | root of two . Right now , of course I | |
| 18:06 | can leave it like this , but I can simplify | |
| 18:08 | further because you know that when you have three times | |
| 18:10 | squared of two , that basically you can flip the | |
| 18:13 | order of these guys here , Right ? So if | |
| 18:15 | I wanted to , I could say uh four to | |
| 18:18 | the third power raise to the square of to see | |
| 18:21 | all I did was take these guys and flip them | |
| 18:23 | around , right ? And I know that four to | |
| 18:25 | the third power is four times four times 4 , | |
| 18:28 | 14 . 4 , 16 , 16 times four Is | |
| 18:31 | 64 raised to the power of the squared of two | |
| 18:34 | . Of course you could leave it like this . | |
| 18:36 | But this is simpler because I've reduced the exponent as | |
| 18:39 | far as it can go . The base comes out | |
| 18:40 | to be 64 . These are exactly equivalent but this | |
| 18:43 | is gonna be a little bit simpler . Yeah . | |
| 18:47 | All right . Let's crank right along . What if | |
| 18:48 | we have three raised to the power of the squared | |
| 18:52 | of two and then we raise that whole thing to | |
| 18:54 | another power of the squared of two . So this | |
| 18:57 | would be like think three squared and then I'll square | |
| 19:00 | that . It's just that the exponents are weird . | |
| 19:02 | They have these radicals in there . So it's still | |
| 19:05 | a power raised to a power . So I can | |
| 19:07 | multiply those powers together so I can do it how | |
| 19:10 | you can write it however you want . But I'm | |
| 19:11 | going to write this is three raised to the square | |
| 19:13 | root of two squared . Now this might look a | |
| 19:16 | little weird to you but when I multiply the powers | |
| 19:18 | together , it's squared of two times the square root | |
| 19:21 | of two . That means squared of two squared all | |
| 19:23 | of this sits in the exponent of three . Now | |
| 19:26 | the two will cancel with the radical because a square | |
| 19:29 | cancels with the square root . That's going to leave | |
| 19:31 | me with just a two . What's left over ? | |
| 19:34 | Basically , you can see that the two here will | |
| 19:36 | cancel with the radical . Leaving a two behind and | |
| 19:39 | three squared is nine . This is the final answer | |
| 19:43 | . All right . Getting some practice here . Not | |
| 19:46 | too bad . Let's crank through the next one . | |
| 19:49 | What if we have three to the power of square | |
| 19:53 | root of two plus two over three ? To the | |
| 19:57 | power of square two to minus two . Now notice | |
| 20:00 | that this plus in this minus . They all exist | |
| 20:03 | in the exponent . This plus two and this minus | |
| 20:05 | two . That's all upstairs in the expo So a | |
| 20:09 | lot of students will look at something like that , | |
| 20:11 | stare at it and just have no idea where to | |
| 20:13 | go . But remember when you have things added together | |
| 20:16 | in an exponent , you can always write it as | |
| 20:18 | a multiplication . These are added together . So this | |
| 20:21 | is exactly the same thing as the square root of | |
| 20:23 | two . And I'm gonna multiply by another three squared | |
| 20:27 | . Why ? Because the basis of the same , | |
| 20:29 | I can add the exponents together and that's exactly what | |
| 20:31 | I have here . So I just break it apart | |
| 20:33 | and go backwards On the bottom . It'll be the | |
| 20:36 | 3 to the squared of two times . You have | |
| 20:39 | a negative 2 , 3 to the negative to power | |
| 20:42 | . All right . So let me kind of rewrite | |
| 20:44 | this here . So I don't kind of destroy what | |
| 20:46 | I have squared of two , three squared and then | |
| 20:50 | I have three to the square root of two and | |
| 20:54 | then three to the negative to now you see what | |
| 20:56 | I have . I have a situation where this entire | |
| 21:00 | number , whatever it comes out to be is on | |
| 21:01 | the top of the fraction but it's also on the | |
| 21:03 | bottom of the fraction so they divide away . And | |
| 21:05 | so what I have is three squared over three to | |
| 21:08 | the negative two . There's several ways you can do | |
| 21:10 | this but I'm gonna write this three squared as just | |
| 21:12 | nine and then this is gonna be 1/3 squared which | |
| 21:17 | is all still in the bottom here . So it's | |
| 21:20 | gonna be 9/1 9th . That's what's down there . | |
| 21:23 | And so I can do the division here . This | |
| 21:25 | will be 9/1 . That's what's in the numerator , | |
| 21:28 | change the fraction division of multiplication , take this and | |
| 21:31 | flip it over . It will be 9/1 and it'll | |
| 21:33 | be 81 . 9 times nine is 81/1 . So | |
| 21:35 | it's just 81 and that's the final answer . Now | |
| 21:38 | . Of course , I could have just done it | |
| 21:40 | from here . I mean there's multiple ways I can | |
| 21:42 | do it . I can subtract these exponents if I | |
| 21:45 | want to , I could have gone , you know | |
| 21:47 | , from this step . Right here I have , | |
| 21:49 | this is gone . So this will be three to | |
| 21:51 | the two minus a negative too . Because I can | |
| 21:55 | subtract the exponents . Oh , that's gonna be three | |
| 21:57 | to the fourth power . And when you do three | |
| 21:59 | to the fourth power , three times three is 99 | |
| 22:02 | times three is 27 . 27 times three is 81 | |
| 22:05 | . So you get exactly the same thing whether you | |
| 22:07 | do it this way or if you kind of go | |
| 22:08 | the other way that I went , you're always going | |
| 22:10 | to get the same answer . All right . So | |
| 22:14 | , same rules of algebra are always applying . All | |
| 22:20 | right , we have a couple more . Mhm . | |
| 22:22 | What if we have ? Okay , Let's take a | |
| 22:26 | look at seven to the power of the square root | |
| 22:29 | of three , Multiplied by 7 to the power of | |
| 22:33 | square root of two . How do you handle that | |
| 22:35 | ? Well , the basis of the same , but | |
| 22:37 | the exponents are different . So I can just add | |
| 22:39 | those seven to the square root of three plus the | |
| 22:42 | square root of two . That's all exist in the | |
| 22:44 | exponent . I can add this is like X squared | |
| 22:46 | times X to the third . I can add the | |
| 22:48 | exponents . Now , a lot of students will try | |
| 22:51 | to do things with this . Well , I'll try | |
| 22:52 | to say maybe this is a square to five , | |
| 22:54 | but we'll add the three and the two or maybe | |
| 22:56 | they'll do something else . Weird . I mean , | |
| 22:57 | the truth is sometimes this is kind of a trick | |
| 23:00 | question and this is one of those . There's nothing | |
| 23:01 | else you can do . You add the exponents , | |
| 23:03 | but you can't really add them because what's underneath the | |
| 23:06 | radical is different . If you remember from radical rules | |
| 23:10 | , you can only really add them if they're like | |
| 23:12 | terms and these are unlike terms . If it was | |
| 23:15 | the square root of three in the square of three | |
| 23:17 | , I could add them . But they're totally different | |
| 23:19 | . So you just have to leave it like this | |
| 23:20 | . So it's a weird expression , 7 to the | |
| 23:22 | power of this stuff . But that's what it is | |
| 23:26 | . All right . So let's change it a little | |
| 23:28 | bit from what I have above . Let's change it | |
| 23:30 | and make it seven to the square root of three | |
| 23:33 | . Raise that whole thing To the square root of | |
| 23:37 | two . So this is the square of three on | |
| 23:39 | the bottom . Sorry about that . Can't really read | |
| 23:42 | that . So let me rewrite it square to three | |
| 23:44 | on the bottom . And then you have a squared | |
| 23:45 | of two here you have a power raised to a | |
| 23:47 | power , so you have the square root of three | |
| 23:50 | times square root of two . Because I'm multiplying the | |
| 23:52 | exponents together . Now . You should remember back from | |
| 23:55 | radicals when you have radicals multiplied , I can just | |
| 23:58 | multiply what's under the radicals ? Seven to the power | |
| 24:01 | of three times two is six . Because when I | |
| 24:04 | have to radicals multiply together , I can multiply what's | |
| 24:07 | under the radicals . So notice that when the radicals | |
| 24:10 | are added together you can't really do anything further with | |
| 24:12 | it . But when they're multiplied together , I can | |
| 24:14 | multiply what's underneath . And that all comes back from | |
| 24:17 | the rules of of radicals . The rules of fractional | |
| 24:21 | exponents . We've done all that in the past . | |
| 24:24 | Okay , So if you have any questions about why | |
| 24:26 | you're doing certain things here , you totally you don't | |
| 24:28 | know why you can multiply these , that all comes | |
| 24:30 | from radical . Uh you know the rules of radicals | |
| 24:32 | who have done in the past . So let's say | |
| 24:35 | you have seven to the square root of three Plus | |
| 24:39 | two , that's in the expo and on the bottom | |
| 24:41 | here , you have 49 . How do we simplify | |
| 24:43 | this ? Well , first of all , you notice | |
| 24:45 | 49 is pretty closely related to seven . So what | |
| 24:49 | I'm gonna do is on the top , I'm gonna | |
| 24:50 | write this is seven times a squared of three . | |
| 24:53 | These are multiplied . So I can multiply . I'm | |
| 24:55 | sorry , added . So this can be times seven | |
| 24:57 | squared . How can I do that ? Because I | |
| 24:59 | can add these exponents together , which is exactly what | |
| 25:02 | I have on the bottom . 49 can be written | |
| 25:05 | as seven times 7 . Right ? So what I | |
| 25:08 | have here is seven to the square root of three | |
| 25:11 | times seven squared over seven squared . And so I | |
| 25:17 | can say well this seven squared cancels with the seven | |
| 25:20 | squared . And so what I will be left with | |
| 25:22 | is seven to the power of square root of three | |
| 25:24 | . And that's the final answer . Yeah . So | |
| 25:27 | I hope you have learned from from doing these problems | |
| 25:31 | that the rules that you apply to exponents when they're | |
| 25:34 | irrational are the same exact rules . Sometimes you can | |
| 25:36 | add the exponents . Sometimes you can subtract multiply and | |
| 25:39 | divide them all . The rules are the same . | |
| 25:41 | There's no difference in the rules but you do have | |
| 25:43 | to get comfortable with the idea of having radicals running | |
| 25:46 | around . It's not just radicals . Pi is irrational | |
| 25:49 | . There are other numbers and math that are irrational | |
| 25:52 | that we'll learn about later that you can have an | |
| 25:53 | exponents but it doesn't really matter . The main thing | |
| 25:57 | for you to understand is that anything you have an | |
| 25:59 | exponent follows the rules of exponential nation that we've learned | |
| 26:02 | . Now . If you want to learn the details | |
| 26:04 | behind how a computer calculates those numbers on the screen | |
| 26:07 | , we kind of went through that , we talked | |
| 26:08 | about the concept of a limit . Where in calculus | |
| 26:11 | you're gonna learn that you have a function . This | |
| 26:13 | is called an exponential function where the variable itself . | |
| 26:16 | This is not like x square X squared is not | |
| 26:19 | an exponential function because this is just a variable race | |
| 26:23 | to a number . This is different , this is | |
| 26:25 | the number raise to a a power that is a | |
| 26:28 | variable that's called an exponential function we can use and | |
| 26:32 | make that exposed to get closer and closer and closer | |
| 26:34 | to the desired value through this thing called the limit | |
| 26:36 | and you get more and more decimals to get more | |
| 26:38 | and more accurate to answer . That's a foundation of | |
| 26:41 | calculus one . Alright , but for now what you | |
| 26:44 | need to know is how to handle these things , | |
| 26:45 | how to handle these irrational , expose , we've done | |
| 26:48 | that here and also to know that this is an | |
| 26:49 | exponential function . We're going to talk extensively about that | |
| 26:52 | probably six or seven or eight lessons and then we | |
| 26:55 | also have the concept of a logarithms which is very | |
| 26:58 | closely related to an exponential function . Also we're gonna | |
| 27:01 | get there very soon . So follow me on to | |
| 27:03 | the next lesson , we're gonna wrap up the concept | |
| 27:05 | of these irrational exponents in algebra . |
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